Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,8)(9,10)(11,12)(13,14)(15,16)(19,23)(20,22)(27,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,16,12)(17,18,21,24)(19,20,23,22)(26,29)(30,32);(2,6)(4,7)(10,14)(12,15)(17,19,21,23)(18,22,24,20)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,15)(14,16)(17,20,21,22)(18,23,24,19)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=g,c^2=fgi,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4605360);